Building Enduring Mathematical Understanding, one lesson at a time.

Archive for the month “August, 2012”

Tech Talk: Keynote + Absolute Board = Awesome!

Continuing my series on iPad Apps. . .

When I first received my iPad a year ago last spring, I was DESPERATE to use it in the classroom. My school laptop was getting old and cranky, freezing up at the most inopportune moments. I have used Power Point in the classroom for quite awhile – not to “tell” students information, but to “ask” them questions. I searched through the App store quite extensively, looking for something cheap to do the job, but nothing looked too promising, so I bit the bullet, forked out $9.99 + tax and bought. . .


My kids use an Apple laptop at home, so I was familiar with the program. (We actually bought a “five-user pack” of Keynote/Pages/Numbers way back when. . . I cringe when I think of how much we spent. Who knew what would happen to the App industry?!) The iPad version doesn’t quite have as many features as the Mac version, but an upgrade sometime during the past year brought the two closer together. I have been able to email my old PowerPoints to myself and open them in Keynote. Occasionally there are a few glitches – borders on a table don’t show up, or a cool font isn’t available – but for the most part it has worked out well. I have different slide backgrounds that I use for different activities in the class, and have added to my collection with the ones in Keynote. Working on the iPad is more enjoyable than on the laptop, and moving or modifying slides is a snap. You can “nest” slides under one another, so this summer I combined all of my slides for each day of a unit together, with a general lesson plan as the “top” slide.


Clicking on the triangle by the slide will “collapse” all of the slides underneath.

Drawback #1

One glaring absence in Keynote is the ability to use superscript (and subscript, but I don’t need that NEARLY as often.) It is really unfathomable to me that this is not available in the font modifications. HOWEVER, I have found a work around. Remember when I said I emailed PowerPoints and opened them within Keynote. The superscript STAYS so I just have a “fall back” slide that I go to when I need to use an exponent. I can change the font/color/size and the superscript will change along with it! (Strange, but true!) I use a similar “shortcut” with square roots, as the keyboard shortcut for the radical symbol is non-existant, but I’ve copied it over from PowerPoint. Since the highest level of math I teach is Algebra, it’s not as if I need a full-fledged Math Editor (although it would be nice!) I am playing around with Mathbot and TeXit and learning a bit of LaTex so that I can paste in some more complex equations later on this year.

Drawback #2

I always though it would be nice to be able to “draw” on the slides! It is, after all, an iPad App! There is really no “freehand tool.” You can created shapes and lines and curves, but you can’t just “scribble something out” on a slide. At first I was really bummed about this, but I have come to realize that maybe I wouldn’t really want to, since I would need to use the slide again the following period. It would be kind of a pain to make multiple copies of each of the slides I wish to write on, so maybe not such a deal-breaker. (Although, it still would be an awesome feature just for CREATING the slides, but I doubt that Apple is listening.) So. . .from the multitude of options that I have downloaded, played with, and even used for awhile, the winner in the end is the FREE App. . .

Absolute Board!

It really IS free!


The way I use this in conjunction with Keynote on my iPad is that I will take a screen shot of the PowerPoint slides I plan on using so that they are stored in the Camera Roll. When I pull up Absolute Board I can quickly pick the slide and it will fill the screen. (For awhile I did it ahead of time, but it doesn’t take any longer to grab the slide than it does to select it from the pages stored in Absolute Board.) I can zoom in and out, change pen size and color, and write down a solution process as a student shares it aloud. (I have found that it is a BIG time saver to have me record rather than a student “write” on the iPad. There are other opportunities in class for them to write.)


The “marked up” slides are then saved in Absolute Board, and I can pull them up later. I am not sure what the limit is on the number of “pages” you can store. Every once in awhile I will “purge” the old drawings, but I can also save them to my photo album if I wish!

These two together make a great combination for me!


Made4Math Monday: Sticks and Seats

One of the benefits of working part time for a NUMBER of years was that I could find the time to be a “parent volunteer” while my own kids were in Elementary School. I must say that it was a valuable experience as a teacher as well. Most of my teaching experience up until that point was at the high school level, so many of the classroom routines were new to me:) One of those ideas was “picking with Popsicle sticks.”


The question is. . .how do you pick sticks when you have six different classes? Do you have six different cups of sticks? After a few years of “refinement” I have a method that works well for me. I picked up a package of big, brightly colored “craft sticks” at the dollar store. There are actually six different colors, but I generally only use four. I wrote the numbers 1-8 on the bottom of the sticks and I am good to go for a class of up to 32 students. Each student is assigned a color and a number (which they remember quite readily after the first few days.) I, on the other hand, have a “cheat sheet” that I post on the board and “borrow” during class so I can actually call on the student’s name instead of just the “color-number combination.”

Here is my “Wizarding World of Harry Potter” butterbeer cup that I will being using this year. (I am finally retiring my University of Minnesota cup with the lettering all worn off, but I am still using it to hold my little six inch rulers.)


Here is one of the (currently empty) “cheat sheets” that will be filled with names once our class lists are finalized.


(Ha! I just noticed the Yellow column twice. I must have used my class that only had three colors last year and copied and pasted a new column. It’s supposed to say Green!)

Soooo. . . what if you have more than 32 students in a class? I’m glad that you asked. There are a number of different ways that you can deal with this issue. Since there are more than four colors available, you can certainly use more colors and fewer sticks of each color if you wish. Last year I had a class of 34 students and I included two blue sticks (in addition to the red, orange, yellow and green ones) numbered 1-2. If you have far fewer than 32 students (less than 24, for instance) you can eliminate one color all together, and never “pick” that color during that class. I have an additional “rule” that I follow as well. If I pick a stick that does not “belong” to anyone in the class (not just because they are absent) then I am the one who must answer the question! (Occasionally I have a class with lots of “unclaimed sticks” and I end up “calling on myself” more often than I wish, so I will put a limit on how many times I can answer in a period.)

I rarely use the sticks for “total cold calls” in class. Quite often students work on a problem, discuss it with a partner, and then I pick a stick to share their response. Other times I will have students work on five or six problems and as I begin to pick sticks, the first person chosen is allowed to decide WHICH of the five or six problems he or she wishes to share. (If no choice is made, I will choose!) By the time the last person is chosen, at least we have already discussed the other problems – even if the most challenging one was left for last. However, this is certainly not always the case. Some students definitely WANT to share their response to the toughest problem! I do not allow students to “pass,” even if they have not completed the problem. I will instead ask questions, questions, and more questions, to help them reach a solution. I will also call on raised hands after a problem has been shared if students wish to add more information, an alternative process, or an alternative solution. I generally leave the stick out of the cup for the rest of the period. I am not sure this is a “good thing.” Some students then “relax” knowing they won’t be called again. Others are disappointed that they won’t get “picked” another time.

I have to chuckle sometimes at the responses I get when I start to pick a stick. (Some times I will have already grabbed it, just waiting for the time to call.) Since they know their color, some are happily anticipating that it might be them, while others are nervously hoping it WON’T be them! Often the people up front will see the number as the stick is drawn (apparently I’m not very adept at hiding it) and actually KNOW the particular student before I can even look it up! Occasionally a student professes “disbelief” because they (let’s be honest, it’s usually a “he!”) “called it” that he would be picked. Oh, sixth graders – I will miss them this year ๐Ÿ˜ฆ


Why all this hassle just to assign a stick to a student? I have ulterior motives. ๐Ÿ™‚ The color-number combination is also often used to assign seats! On a daily basis, students enter the room and need to figure out where they are sitting and who they are sitting with for that particular day. (I alluded to this in my First Day post, but here is the full text version.) Some days the desks will be in groups of four and they might be sitting with their number groups or “half” of their color groups (evens and odds or highs and lows.)

Other days they will be sitting in two’s where they are generally paired up with someone else in their color group.

Since there are usually six or seven other people in their color group, this partner will also change from day to day. Part of my reason for posting the lists is that they act as a “cheat sheet” for the students as well.


You may have noticed signs above the class listings that notify students the groups for the day. I have laminated (double-sided so I can just flip for a new option) all of the different seating choices available. For the “color pairs” there are seven different signs that all have each number paired with a different “partner.” (Now there’s a math problem for you!) I also have laminated signs for each number group that I set out on the groups of four, and for each color group that I place out to determine the “row” or “section” for that color. These are all stored in a basket right under the section of the board I use for group assignments.


Just to make things even MORE confusing, I also have a few additional ways of picking the groups for the day. One is “Find Your Match” (pairs or trios) that I described in my post on Math Cards. The other is “Just for Today” groups (pairs, trios, or quads.) I often use this option after a formative assessment or during review activities where I purposely group students so that at least one person in the group has a strong understanding of the concepts. I sort their formative assessments and put them in groups, then jot down the names on a blank seating chart in a page protector using an overhead pen (I guess they still have their uses!)


I assign new “color groups” every quarter. (I used to do it more frequently, but it can be time consuming, and with eight people in group, plus the other options, they get quite a variety throughout the quarter.) Initially the groups are usually alphabetical. By the second quarter I put some work into assigning groups. I often have the highest performing students with the same number (or two to three numbers) so that I can choose to use “Number Groups” when I want to differentiate a bit. (Those groups would “receive” a more challenging problem than some of the other groups.) I am also aware of when I end up with “high-low” pairings so that I either take advantage of built in “tutors” or at least I do not plan an activity in which partners might be “competing” against each other. A definite part of my lesson planning involves deciding how students will be grouped for the day. Finally, for the last quarter I usually allow some student input regarding who they would like to have in their group. I take requests of 2-3 people for each student and I can usually place them all in a group with at least ONE of the people they requested, often with two (or another in their number group.) Again, THIS is a challenge as well!

Back to Sticks

There were blue sticks I used in my class of 34 last year. When in “Number Groups,” they just created a group of five. When in color groups pairs they were my “Wild Cards.” They took the place of students who were absent for the day or sat together as a pair. (I also randomly drew sticks that they would switch with so each day there were different student’s sitting in the “Wild Card” seats.) For odds/evens or high/lows, if everyone was present, I would have them join a “convenient group” with the most extra space to form groups of five. I would also probably do this for a class of 25 or 26 instead of having four color groups with “lots of empty seats.”

Last year I came up with a new way to use the sticks during group work. If the students were in Number Groups, I would walk around with one stick of each color in my hand. When stopping to check on a group or to answer a question, I would place the sticks behind my back, mix them up and pick one to determine who to call on to either ask or answer a question. If students were in Color Groups, a collection of sticks with different numbers could be used. (Although I also used an octahedron, but this sometimes resulted in MANY rolls if I was at an “even” group and the numbers kept coming up “odd.”)

Whatever For!?

Sticks: I am TERRIBLE at calling on “purely random” students, so the sticks help me to do that on a more regular basis. (We also have class discussions that don’t involve sticks – it depends on the particular activity.)

Seats: I want every student in the class to be comfortable and familiar working with every other student in the class. We all have our strengths and weaknesses that we bring to a group and I want student to be aware of that fact, and looking for opportunities to share their strengths while acknowledging and working on their weaknesses. Not every student “enjoys” working with every other student in the class, but they know it will change the next day. Sometimes we will stay in the same groups for two consecutive days, but really no more than that. Some students swear to me that they have been with the same partner waaaay too often because they “happen” to end up partners in find your match (over and over. . .), they are IN the same color group so they are in pairs and quads with that person, AND I even put them together in a “Just for Today” group!! Oh, the injustices of being a middle school student!

Minor Pitfalls

Last year my sixth graders were my first class of the day. The buses arrive by 7:30 and class starts at 8:00, so more than a few of them would “hang out” in class for awhile. Then, I started noticing some patterns. Within a “column” of eight seats, the pairs can choose their spots on a first come – serve basis. Some students would quickly “claim” spots so that they could be just across the aisle from their “BFF” – who they were not actually in a group with (on purpose, from my point of view) but wished to sit near them anyways. I began to be quite careful about where each color groups was assigned, or where each number group was placed to try to avoid the “cliques.” I was not always successful. This year I have 8th grade Algebra students first thing in the morning. I am not sure they will “hang out” in class before school, but if so, I think I will wait until closer to 7:55 to make the group placements for the day!

Whew! “Leadership Team” meeting this morning, working in my room all afternoon, and I still finished this post at a decent hour – West Coast Time! Still nine more days ’till students arrive – unless you count out Open House on Wednesday, but I think I’m ready for that ๐Ÿ™‚

Reflections: Lurking to Learning

Today is my one month “blogversary!” I don’t have much to add to the conversation on Advisory for msSunFun, but I do have an item on my To Do page that I would like to tackle: My journey to becoming a blogger.

I first “lurked” onto the “mathedublogger” scene just over SEVEN years ago! At the time I was teaching at an Alternative High School, and we had just been “chosen” to participate in a 1-1 program beginning during the 2005-6 school year. We (the teachers) had some training at the end of the school year and then took our laptops home over the summer. Wow! The Internet in your lap! During the hours/days/weeks spent searching for online tools and resources I ran across some “bloggers” talking about what they do in the classroom – especially in regards to technology. It was still a fairly new idea from my perspective. Every once in awhile I would wander back and take a look at what was happening. Often one site would link me to another, and so on, and so on. . .

Meanwhile, I moved to one of the middle schools in the district and was no longer involved in the 1-1 program. However, our Computer Lab teacher shared GoogleReader with us, and my lurking seriously went up a notch or two. My tiny iPod touch became my window into the “mathedubloggosphere.” I don’t have the data to back it up, but I really do believe that the MathEd blog scene has grown exponentially – meaning that the growth was actually quite gradual to begin with, but then started to take off! I remember running across dy/dan (what a cool blog name, I thought) before he WAS dy/dan! I “lurked” as some of my favorites, Continuous Everywhere but Differentiable Nowhere, f(t), and MathMamaWrites built their followings. ThinkThankThunk next slammed onto the scene along with his SBG cohort Point of Inflection. Even though most of the bloggers were teaching at the high school level (or higher) and I was now working with sixth graders, I could relate to a lot of what they were saying – especially in regards to SBG. I thought, hmmmmm, maybe I should do this. However, I am the LAST one in a large group of people that I don’t really know to actually speak up. . .And I wonder why my own kids are so shy. . .

The next “jump” in my lurking occurred when I got my iPad and found Flipboard, that I shared in this post. It makes scrolling through blogposts pure pleasure! I would look at the blog rolls of the bloggers I was reading, check out new writers, subscribe to their posts, and so on, and so on. . .

After returning from our vacation this summer, I read some posts about Twitter Math Camp. Seriously? These people just planned their own “retreat/workshop/conference!?” Wow! More new bloggers were added to my feed. . .and then one afternoon I was weeding out in the yard, listening to my iPod when I heard the lyrics, “This is your life, are you who you want to be. . .” and something just clicked. Yeah, I can do this! You know, it makes so much sense to do it NOW, when I am just going back to teaching full time after being part time for 16 years. Sure, I have PLENTY of time n my hands! Oh well, I downloaded the WordPress app (recommended by Sam) and signed up for Twitter to boot! After all, I had already picked out a name and everything. The rest, as they say, is history. . .but not really what I expected.

It has been a roller coaster ride over the past month. I started out with ideas about what I wanted to “say,” but I find myself “saying” a lot of other things too. msSunFun came onto the scene shortly after I first started. “Ok, I’ll try that.” Then I noticed the Made4Math Monday. “Suuuure, that too.” Sam posted his New Blogger Initiative. “Hey, that’s me! I’ll sign up.” All of a sudden I have more ideas to share than there are hours in the day. How to keep up? Oh yeah, and school is starting up soon, too! My Start, Stop, Continue post includes blogging goals that I hope I can keep. I even made a To Do’s page to keep myself honest.

Posting comments on other blogs was one of my first “baby steps” after I started my blog. I really appreciated it when the blogger would then reply back. Especially now that I added all of the NBI bloggers to my feed, I find myself overwhelmed with how much there is to read. I want to make “meaningful comments,” as opposed to “I really love that idea.” Maybe that’s more of a “Twitter” response to a post.

Twitter has been hard to “jump into.” I once tweeted that I felt like the “new kid at school,” just listening in on other conversations – except they may have happened hours ago! It is very strange sometimes. I will “reply” and then realize that the person may have already moved waaaay beyond that part of the conversation. I don’t know how people can even BEGIN to follow as many people as they do!

When I was lurking it was so much more of a passive experience. Now that I am “in it,” I have been learning sooooo much more from others. I am reading posts and tweets from Middle School Math teachers who are out there in force, (@jruelbach, @4mulafun, @fawnpnguyen, @mr_stadel, @Borschtwithanna, @mathbratt, the list goes on, and on, and on) and I didn’t really know about them before. I LOVE the posts and twitter conversations with/between those involved in Math Education (@delta_dc, @mathhombre, @ChrisHunter36, @trianglemancsd) that make me think more deeply about learning mathematics! I feel a kinship with other “newbies” like me (@danbowdoin – although he is on the fast track to blogger stardom, @G8rAli – who should really start a blog, @aekland – who has such thoughtful posts, @ray_emily – who has an abundance of enthusiasm, and Pai Intersect – who I haven’t seen on Twitter, but has great insights on his blog.)

I vacillate between thinking that “nothing I have to say has any value when compared to all of the ideas that others have shared” and “oh, I really want to chime in,” or “I think I should share that. . .”

I am surprised at how much I have learned about myself and my teaching from writing posts for the blog. @ray_emily tweeted earlier today: “I’m finding I have a new clarity / fresh eyes on a topic after blogging about it.” I entirely agree! I am especially looking forward to learning even more as I blog about my experiences in the classroom ๐Ÿ™‚

New Blogger Initiative: Integer Context Cards

For Week 2 of the “New Math Blogger Initiative” (or is it an “initiation?” hmmmmm) I decided to learn how to embed a document using Scribd and show something that I am proud to share!

If you read my Made4Math post from last week (Math Cards) you know I like “multi-purpose” tools. I haven’t been teaching Middle School too long, in the grand scheme of things, and when I first had the opportunity to introduce Integers to a class of very low sixth graders (all Level 1 on the state test) I knew that putting them into context would make all the difference. So. . . what are some contexts for integers? Well, there’s temperature, and altitude, and money. . .? I brainstormed long and hard and came up with quite a list.

Without further ado, my very first Scribd document!

Integer Context Cards

(Hmmmm. Rats! The font changed when I uploaded the document. The original was in Herculanum, a very cool looking ALL CAPS font, so now it appears that I don’t know my capitalization rules – oh well.)

There are a total of eighteen different contexts with six cards for each one. Some are “stretching it” a bit, but still reasonable. Note: for altitude, one of the cards is for Arlington, Washington where my school is located, so you might want to “personalize” that one. ๐Ÿ™‚

I made one copy of each page on a different color of copy paper to make them easier to sort and then had them laminated, cut, and paperclipped in sets of six. (I only made one set for the entire class, but you can certainly create duplicates, especially if you have a very large class.)

Order, Order, Order

Phase 1: After a brief exposure to a few of the cards, we started our first activity with the cards. My students were already assigned to groups of six. We actually went out into a space in the hall for this.) Each group received a different set of cards, passed them out amongst the group members, and silently “raced” to put themselves in order from least to greatest. Once a group was done, they ALL had to raise their hands. After I checked for accuracy, they turned in their set and grabbed a new one. (The first year I did this I checked the groups off on a master sheet, but the following years I just trusted their memory – “We already did that set.”) The goal was to accurately get through as many sets as they could in the time allotted. Often enough, groups were in too much of a hurry to read carefully enough to identify the “key words” that signified whether their value was positive or negative. Getting a “no” when their hands were raised definitely encouraged them to take their time a bit more.

Phase 2: The next activity took it just a bit further. Groups (of three this time) had a sheet of number lines. Each time they received a new set of cards they had to “fairly accurately” plot and label all six values (along with zero if it wasn’t in the set) on a number line. Choosing a scale was challenging for some of the sets (especially with the first group of students.)


(See the cool font? Oh, well.)

Integer Operations in Context

A few weeks later in the year, my seventh grade class was working on operations with integers. I printed the Integer Card sheets four to a page and created little packets for students to share. (See the image above.) Using a “Think Pair Share” type of model, I posed questions using the people, places, or things on the cards and students had to write a math sentence, model the problem on a number line, and find the value (first on their own paper, then with a partner on a mini-whiteboard.) A big key was writing the math sentence as opposed to just finding the value. I wanted them to make the connection so that when they saw a “naked numbers” problem they could try to connect it to the contexts we worked with in class.

We initially focused on addition and subtraction situations:
(**The white text is shown first. After the Think-Pair-Share on whiteboards I reveal the number sentence and diagram and move onto the next problem.**)

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Next we moved onto subtraction in “finding differences” contexts, as well as multiplication and division. I dropped the “number line” requirement, although we did end up sharing it on some of the more challenging multiplication/division situations. (Again, the white text for each problem is given first, TPS, then the yellow answer is revealed, discussed, and we move on to the next problem.)

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In subsequent years I used the Integer Operation activities with my sixth grade classes as we introduced this seventh grade concept in the Spring after the state test. During “Review Activities” time in class I had small groups create original “stories” along with their associated number sentences, from the cards, but I think I would like to make that a more integral part of the initial learning as well.

Ahhhhh, context! Even if it is really “pseudo context,” in this situation students have something to grab onto that helps them make sense of the integer operations as opposed to a mercurial “set of rules to follow.” On the other hand, it is THROUGH these experiences that students begin to create their OWN rules about the patterns involved in integer operations, but only when the concepts “make sense” to THEM! I think we are finding a bit of EMU right there. ๐Ÿ™‚

Note: There are multiple slides for each operation, and if I got my act together I might copy and paste them all into one slide show and try to attach them, but that would take a higher level of understanding on Scribd than I currently possess, as the slides are all now on my iPad, and I am just sharing screenshots.

Made4Math Monday: Monster Whiteboards


Yesterday I finished getting my “monster” whiteboards ready and I brought them into my classroom today. I am far from the first to have created these learning tools. Frank Nochese sang their praises here, and Anna followed up with another post as well. I use the little mini-whiteboards quite regularly with pairs, and they will still have a place in my class, but the opportunity for groups of three or four is really exciting! I especially envision using them for problem solving, such as the chessboard problems for my First Day Activities. I am concerned that we will not always have enough time to complete the “solving” as well as the “sharing” in one class period, but my solution will be to at least take photos of all of the boards and project them on the screen the following day as groups present. Another intriguing use will be the Mistake Game, as described by Kelly O’Shea. (Go there! Read it!) Sooooooo looking forward to trying this out – especially with my Algebra students because I think they will thrive on it. However, in the long run I predict it will be incredibly valuable to my Math 8 students in freeing them from fear of failure. The classes on the whole are not full of students who have been successful in the past, and developing a classroom culture where mistakes are acceptable and even celebrated as ways to learn concepts more deeply (EMU!) will be a huge step for their learning.

I bought mine at the local Home Depot for about $13.50 per sheet. I picked up two that they cut (for free!) into the six 24″ x 32″ pieces recommended by Frank. (I considered going 2′ x 2′, but I am very glad I went with the extra inches on the length.)

My next task was to put duct tape (or duck tape) on the edges to keep them from degrading. This is where Anna gets all the credit! I had planned on “buying” some of my daughter’s stash


until I “did the math!” Each board requires almost ten feet of tape (112 inches of perimeter, plus a bit extra on either end that I cut off while taping.) Since I had twelve boards, the 120 feet of tape would have decimated her supply. (As it was she wasn’t going to give me the “fancy” stuff anyways.) I decided to go with my school colors and picked up a blue roll and a yellow roll of 20 yds each. Needless to say, there’s not much left. (Maybe just enough to use for periodic “repairs.”) The tape cost $7 for the two rolls, for a grand total of $34 + tax, or about $3 a board. (The cuts were free, and didn’t take long at all, but the taping process took me over an hour!)

Here they are!


Notice as well, the PERFECT storage space that is holding the other ten!

Now, for the writing instruments. I am always frustrated by how quickly the dry erase markers die. Last year I picked up some of the Crayola Dry Erase Crayons. Other than the periodic breakage, they seem to “last” much longer. (Hey, when they break, one crayon turns into two!) In addition, I have to watch out for pieces that accidentally “chip” off, are left on the floor, and then get ground into the carpet.๐Ÿ˜ž The crayons look pretty sharp (sharp = awesome, not sharp = pointed) on these boards, and the color variety is great, with the exception of the yellow crayon – whoever thought it would work well on a whiteboard was sadly mistaken. I guess they sell yellow markers as well – maybe it’s for those black dry erase surfaces. I “inherited” a number of boxes of the crayons (eight colors in a box plus an eraser “mitt” and sharpener) when I moved into this classroom – enough that each group could have their own box! (I will generally only use eight boards at one time.) The kids liked markers better, but maybe the different colors will “sell it.”

One more note about the duct tape. In hindsight I don’t think yellow was a great choice. As I was erasing one of the boards today I realized the dry erase particles will accumulate along the edges of the tape, so lighter colors will start to look “grungy” after awhile. Oh, well.

One “short” story about my whiteboard history. In the early nineties (yes. . .I am dating myself, but I have already done that on a previous post) I was teaching Calculus. I had one particular student who repeatedly asked if he could go up and work out problems on the corner of the board while the class was working on a set of problems. It was certainly fine with me. He commented on how his thoughts seemed to “flow” from his brain when he used a whiteboard. It was not long before he bought himself a 1′ x 2′ framed board to use at home, as well as another one that he brought to school (and stored in the classroom) to use at his desk. I have no idea where he found these boards, as I had never seen them in stores. At the end of the year he gave me the one he had brought to school and I still have it to this day. (It was one of the last years that I taught Calculus. Not many years later, I went on maternity leave and then came back to teaching just part time. My own kids scribbled and doodled on it while they were toddlers.) Jump forward about ten years, when I was volunteering in my son’s first grade class, I observed his teacher using a class set of mini-whiteboards with her students. Within about a year, I had a set for my classes. Now I am on to the next phase – MONSTER whiteboards!

(Ok, it wasn’t so short. Have I mentioned that I ramble . . .?)

msSunFun: Start, Stop, Continue


For this week’s msSunFun, I will have to opt for the goal-setting post, as my room is still somewhat in chaos. (I “inherited” a room from a retiring teacher and he left me lots of “gifts.” It took hours to sort through and decide what I wanted to keep, so now I am finally making some headway. We don’t actually start with students until September 5, but we do have an Open House a week from Tuesday – yikes!) After using 12 different rooms over the past 17 years, sharing with another teacher in all but one, am I hopeful that I have found a “home” for awhile!

Soooooo, on to the “real post.” Not much discussion – just a list. (I rambled on far too long yesterday for the New Blogger Initiative post.)


-Blogging about one lesson (success or “failure”) each week to share/get feedback.

-Incorporating the use of “monster” whiteboards in group problem solving.

-Implementing ideas I read about on blogs/twitter by figuring out how they will work best in my classroom.

-Introducing other teachers in my school/district to some of the ideas I read about.


-Staying up too late and/or falling asleep while reading with my daughter and then not being able to fall asleep later.

-Thinking critically about teachers who teach in an “ultra-traditional” manner and try to look for ways to help them see things differently. (Ok, that’s a stop/start.)

-Putting off following through on incomplete/missing student work. Help students understand how much I value homework even if it isn’t part of their grade. (Another stop/start.) ๐Ÿ˜‰

-Allowing “little things” to disrupt the learning environment without addressing them in a simple non-threatening way.

-Applying too much pressure on myself to be the “perfect teacher.”


-Trying to focus on EMU (Enduring Mathematical Understanding,) especially when first developing concepts in the classroom.

-Using SBG to help students (as well as myself) see where they are really at in terms of their learning.

-Thinking “outside the book” when it comes to planning lessons and activities in the classroom.

-Writing posts for the New Blogger Initiative and msSunFun on a fairly regular basis (depending on the topic) and throw in a little Made4Math Monday and My Favorite Friday for good measure.

-Reading all the wonderful ideas that others are willing to share while attempting to find time to comment on them as well!

I think that’s all for now. I’ll check back in about a month and see. . .

Reflections: Why “findingEMU?”

There is a blurb on my “about” page that describes the title of my blog:

Enduring Mathematical Understanding is found within, built upon your own foundation, framed by your current perceptions, and constructed from your experiences. By sharing my thoughts, ideas, and ramblings with others, I hope to encourage the growth of positive dialogue towards this lifelong goal for myself and others.

Wow! I really came up with that on my “first day” blogging, what seems like ages but was just over three weeks ago! (I did say “ramblings,” and that’s sure the truth.)

I have thought about writing a blog for quite awhile now. (Still to come – my “From Lurking to Learning” post.) Over time I toyed with different choices:

“MathMom” because I was a part time teacher / part time stay at home mom. My kids were (ARE) both math fanatics, and I would like to share some stories about watching them grow as mathematicians. (While I don’t exactly have their permission, I have put a few memories down on “paper” over the past few weeks.) However, “Math Mama Writes,” a blog that I have read, appreciated, and enjoyed for many years now, already had a claim on that type of title, so I decided to move on. (In addition, my kids have grown and I don’t find nearly as any opportunities to “be” a Math Mom anymore ๐Ÿ˜ฆ .)

My next thought was “Making Math Meaningful” (shortened to M^3….ooh!) While it has a nice ring to it, and it IS a part of what I try to do in my classroom, what does “meaningful” mean? Definition: full of meaning (duh!) significance, purpose, or value; purposeful (you mean, like “full of purpose”?); significant. I DO want to help students find meaning in math, see its value, know that it has purpose. However, my lessons are not “chock full of 3-Acts,” so it didn’t seem to quite capture the “spirit” of my classroom.

Recently, the phrase “Enduring Understandings” seems to have taken its place alongside “Standards,” or “Essential Learnings,” but I really like the use of the term “Enduring.” Teachers complain all the time about lack of retention – students can learn something for a week or a month, but it is no where to be found after that. What is it that makes the learning endure?

Now, “Understanding,” that’s a whole ‘nother can of worms. It seems like last spring, but it was only in June that I read Richard Skemp’s paper on Relational Understanding and Instrumental Understanding. I ran across it by linking from a Math Mama Writes post to The Republic of Math blog. (I can’t imagine why I didn’t read it when it first came out – oh yes, I was still in Elementary School!) It is a powerful piece, written 36 years ago, that, for me, emphasizes the difference between when a student says, “I understand!” and when they truly do. “Conceptual Understanding” vs “Procedural or Algorithmic Understanding” are more commonly used today, but I feel that, in reality only “Conceptual” actually reflects true “Understanding.”

So, that’s what it all boils down to for me. How DO I help my students FIND Enduring Mathematical Understanding? I can’t find it FOR them – sometimes I have a hard enough time finding it for ME. I don’t want the “I can do it!” math, I want the deep comprehension down in their core that builds and branches off from what they DO really know and understand. What I do in the classroom: the questions I ask, the answers I “don’t give,” but the way in which I respond to questions, the student dialogue I promote, the activities I provide that lead without “dragging them along,” and the culture of the classroom that I help establish, are all ways that I can help them on their journey.

I love the crafty stuff, notebook organization, games and activities, and “monster” whiteboarding plans (tooooooo Monday links to post!) that I have been bombarded with over the past few weeks, (I think I have at LEAST tripled my Google Reader) and the discussions about SBG, (or sbar?) how to deal with the homework hassles, and uses for technology on Twitter. However, what I really cherish from the past three weeks are the conversations about teaching fraction division with/for understanding (“I hate “magic math”,”) introducing integer operations with meaning (“I abhor – is that too strong a word? – the “follow the rules” approach”, and re-defining slope (NOT just “rise over run” – KA), running across David Coffey’s ideas about flipping homework that gives “practice problems” an entirely different level of meaning from back in February, and finding Michael Pershan’s fabulous new (in addition to his insightful “regular” one) blog, Math Mistakes where readers are faced with the tasks of identifying not only “what don’t they know,” but what DO they understand and thoughts on remediation – how can I meet them where they’re at?

Deep breath. One sentence?! I’m not editing. (Except to add MORE words, I suppose.)

findingEMU – oh, like “Finding Nemo!” Well, yeah, I went for the “catchy” goofy title that resembles a wonderful little Pixar movie, but with an odd looking flightless bird instead of an adorable clownfish.

Only I guess I really didn’t, because I really AM trying to find ways to reach EMU.

Tech Talk: Fun Free Apps #1

I’m not so sure how these “Tech Talk” posts fit into my “Finding Enduring Mathematical Understanding” theme, but I will stick with them for now. I am, after all, a closet AppleFanGirl with a Master’s in Educational Technology. (O.k., so the Master’s is almost twenty years old. . .not exactly “useful” – except on pay day.)

If you have an iPad, iPhone, or (newish) iPod Touch, and you read ANY blogs at all, you MUST get this App. (Oh, it’s probably available on Android as well.) It is, amazingly, FREE! Soon, you too will be swiping through your Google Reader with ease. The magazine-like interface is so relaxing. Each post shows the title, an image (when available,) and the first few lines of text. One click and the whole post appears. Another click and you are taken directly to the website. Double click and you are back to to swiping pages.


In addition to Google Reader, you can set it up to show your Twitter feed. Images of pictures and links show up right on the page. A few clicks and you can reply, re-tweet, favorite…


Facebook is another social networking option. Flipboard also “curates” posts in a large variety of categories. (See Tech, Cool, and Sports in image.) A few months before the Olympics, I added one on the Olympics. (Maybe this wasn’t such a good choice, as I too often knew the results prior to watching the broadcast – what a surprise!)


I recently, refocused my Google Reader so that it only contains education related blogs, but then I am able to add some other blogs that I enjoy (kxcd, This is Indexed, GraphJam. . .ok, they are still kind of “mathy”) to Flipboard so that I can read them at my leisure. Originally you could only fill one “page” but now it might be unlimited. I’m on my third page as this point.

It doesn’t take much time at all to set up and then you can change things around add/delete whenever you want. It is DEFINITELY worth it ๐Ÿ™‚

Made4Math: Math Cards

When I was taking Math Ed. classes in college we had to choose one manipulative and describe all of the different ways they could be used in the classroom. I chose Legos, and came up with quite a list of activities. Many of them involved using Legos as replacements for other manipulatives, therefore allowing you to spend less by “just” buying Legos. I guess I was frugal even back then ๐Ÿ™‚ (Of course, that was before I knew just how EXPENSIVE Legos are!) Almost all of the Legos at our house are “claimed” by my son, and those that are left belong to his sister, so I have yet to use them in the classroom. (I did “sneak” some Duplos to school when they were younger, but those are all now in storage.)

Anyways, I originally developed the idea of “Math Cards” just for the “Find Your Match” activity listed below, but it has morphed into much more. See what other ideas you can think of!

Find Your Match
The first incarnation of Math Cards involved having students pick a slip from a bucket when they arrive in class and find their partner for the day based on their card. Now, the cards are not identical (this is middle school math, after all) but “match” in some way. Maybe they are equivalent. Maybe one card is the solution to the equation on another. Maybe they are different representations of the same linear relationship. The sky’s the limit on what you can create. Here are some I have made for this year.



I also have “Find Your Match Trios” cards where groups of three will “find each other.”


Each year, I would print out the sheet and cut it up for the class to use. I would create new “cards” for different concepts to add to my collection. (I have quite an assortment for 6th Grade, where we have a two period block with the students.) The next year I would find the file, print it out, cut it up . . .
I found that students would “cheat” (in my opinion) by putting their “answer” on the back of their slip and THEN finding their match, but in reality the “game” involves NO talking, and students are holding up their card and looking at other cards (analyzing them mentally) to find their match. The idea is that they will have to “do the math” for every other student’s card until they find their match.

Sooo, my brilliant idea was, hey, I can have these laminated and reuse them for years and they also won’t write on them! I took it up a notch and glued them on construction papers before having them laminated. Color coding made them easier to keep in sets. I had a TA first semester last year and it kept her busy quite often. Here are some of the finished products:



THEN, other uses started flowing into my brain. Below are “two more ways” that I have been using the cards, but within those two categories are MANY more uses ๐Ÿ™‚

Math Greetings
I have mentioned this briefly in other posts: First Day and Magnets, and I plan to do a more in depth post on it in the future. As students enter the room, they are required to “do math” in some way. I can put a set (or part of a set) in a “bucket,” have a student draw a card and “do it.” Sometimes it means solving the equation, sometimes it means changing the fraction to a decimal, sometimes it means finding the slope for the near relationship, sometimes it means just doing the arithmetic on the card. Again, the possibilities are endless.

Some cards are just numerical values (fractions, decimals, percents, square roots, cube roots) so I combine the cards with student magnets and they plot the values on a number line.

I have also combined the cards with dice.
-Draw a card with an algebraic expression, roll a die, evaluate the expression for that value of the variable.
-Draw a card with an algebraic expression, roll a die, set the expression equal to the value and solve.
-Draw a card with a fraction, roll a die, multiply (or divide!)

This year I plan to add some more “twists.”
-Set out a 4 by 4 grid of cards. Students find a match, I take the cards and place out two new ends.
-Pick two “ax + b” cards: set equal and solve, just add them together, or multiply together.

I really value the brief 1-on-1 with each student. It allows me a quick assessment of where they are at on a particular skill or concept. Often 2-4 students are “answering” their Math Greeting at the same time, so it really keeps me on my toes! Sometimes it really opens my eyes as to the general level of understanding that remained after 23 hours, and this may alter my plans for the day ๐Ÿ™‚

Review Games
At the end of each Unit, I often have a variety of different activities that rotate among groups of students. Many of these end up involving the cards in some way.
-Math Race: Flip a card from a stack, race to answer it, keep the card if you “win” OR everyone writes down their answer, earn one point for getting it right and one for being first (IF you are correct.)
-Match Game: Lay out cards upside down. Flip pairs and try to find a match.
-Go Fish: Students play the game while “fishing” for cards that match the ends in their hands.
ANY of the Math Greetings activities can be modified slightly as well.

Final Thoughts
Most of the time I am quite intentional in the problems I choose. For instance, for the scientific notation set in the first image, there are only two different mantissas (had to look that up) so plenty of opportunities to demonstrate conceptual errors. The Exponent Rules set provides for the same “opportunity for error.” Does 2^20 รท 2^5 = 2^4? Students who think so will “find the wrong match.” However, in the factoring/multiplying set, I could have done a better job just switching two values so that the “a and c” in the quadratic are the same. I once turned my brain into mush coming up with sets of five values where each card matched with two other cards having the same mean, but also matched with two different cards having the same median. It made for a good discussion of outliers!

Once I figure out the “tech side of things” I do plan to post links to them on a page here on this blog, but for now, feel free to come with your own ideas!

Part of Enduring Mathematical Understanding comes through practice, and Math Cards certainly provide that opportunity. However, deeper understanding comes from dialogue. “Find Your Match” is often followed up by questions regarding “why” two particular cards are a match. It’s always interesting to hear the conclusions!

MS Sunday Funday: “My Own Math” Notebooks

For the past few years I have required my students to have either a spiral notebook, a composition notebook, or a three-pronged pocket folder filled with paper. I have provided them one if they do not have one (finding the penny or dime deals) but I have not collected many from the sales this summer. I do have some useable three-pronged folders to hand out for those in need. Most students end up using more than one during the year, but I do not have them “start over” each quarter or semester. They just start a new one whenever the finish one off. (I had some sixth grade girls that wrote microscopically last year and were able to finish the year using only one notebook, but that is the exception.)

My notebooks are not so much “interactive,” but EVERYTHING goes in their notebook, with the exception of assessments (which are stored in class, in their math portfolios.) Students record daily learning targets, vocabulary, responses to class activities, partner activities, writing prompts, whiteboard activities (paper and pencil first), practice problems, homework, and I am sure some other things I left off the list. We also do activities that are entirely hands on and/or verbal. (They are usually quite excited when I tell them they DON’T have to write it down.) Each day they record the date along with “Today’s Targets” and start where they left off. I do have handouts, etc. for them to “attach” to their notebooks. I have used both glue sticks and mini-staplers in the past, but I “inherited” a bunch of rolls of masking tape from a retiring teacher, and I think I have found a way to make good use of that gift!

There are a few new features I plan to add this year. Unless students have the three-pronged pocket folder, I will be giving them a piece cut from an old file folder (another “inheritance”) to tape on the inside of the front and back covers. This will serve to hold handouts temporarily if there is not enough time to attach them in class that day. It will also store a few “ongoing handouts” that I use for formative assessments. These are one page “booklets” (folded the hamburger way) that have space for eight responses. Students will turn them in on days we use them and receive them back with feedback the next day to store for the next use. Another item to be stored in their “pocket” is a homework log on which they record the time spent on homework each day along with a parent signature. I will collect these weekly and return each Monday.

I also am working on designing a generic “foldable” for vocabulary (“Words of the Day.”) I will probably modify the Frayer Model that has been mentioned by others, as my vocab slides include definitions, examples, non-examples and use in a sentence. I do like the idea of including “characteristics” as well. The “tri-fold” below allows either the word OR the definition to be folded on top, so students can quiz themselves either way.


Students will store “blanks” in their “pockets” and pull one out whenever we need them.

Last year I collected notebooks every few weeks. (I had fewer students than I will this year.) I am not sure it was worth the time I spent creating a checklist and paging through them. I use Standards Based Grading, so their notebook is not actually part of their grade. It is important for students to understand the idea that the value in the notebook is in the thinking and recording that happens as they create the notebook, and the resource it becomes for them as they progress through math, not the “points” they earn. Some students need gentle reminders to stay focused, record their ideas, and show their work, whether I collect their notebook or not. The physical and verbal cues I give them in class do more than written comments at the end if the week. We will see how well it works if I don’t collect them this year. ๐Ÿ™‚

I guess that last paragraph sums up the Enduring Mathematical Understanding that I want them to learn from using notebooks. The math they learn is THEIR OWN. They are in control of their own thinking, their own ideas, and their own strategies, as well as what they gain from others. Their notebook is a place to store all that they learn!

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